Michelle Arroyo

2021-11-18

Find the indefinite integral.
$\int \frac{{x}^{3}-4{x}^{2}-4x+20}{{x}^{2}-5}dx$

### Answer & Explanation

David Tyson

Step 1
find the indefinite integral.
$\int \frac{{x}^{3}-4{x}^{2}-4x+20}{{x}^{2}-5}dx$
Step 2
$I=\int \frac{{x}^{3}-4{x}^{2}-4x+20}{{x}^{2}-5}dx$
$=\int \left(\frac{x}{{x}^{2}-5}+x-4\right)dx$
$=\int \frac{x}{{x}^{2}-5}dx+\int xdx-\int 4dx$
$=\frac{1}{2}\mathrm{log}\left({x}^{2}-5\right)+\frac{{x}^{2}}{2}-4x+c$

Ruth Phillips

Step 1: Polynomial Division
$\int x-4+\frac{x}{{x}^{2}-5}dx$
Step 2: Use Sum Rule: $\int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$.
$\int x-4dx+\int \frac{x}{{x}^{2}-5}dx$
Step 3: Use Power Rule: $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$.
$\frac{{x}^{2}}{2}-4x+\int \frac{x}{{x}^{2}-5}dx$
Step 4: Use Integration by Substitution on $\int \frac{x}{{x}^{2}-5}dx$.
Let
Step 5: Using u and du above, rewrite $\int \frac{x}{{x}^{2}-5}dx$.
$\int \frac{1}{2u}du$
Step 6: Use Constant Factor Rule: $\int cf\left(x\right)dx=c\int f\left(x\right)dx$.
$\frac{1}{2}\int \frac{1}{u}du$
Step 7: The derivative of .
$\frac{\mathrm{ln}u}{2}$
Step 8: Substitute $u={x}^{2}-5$ back into the original integral.
$\frac{\mathrm{ln}\left({x}^{2}-5\right)}{2}$
Step 9: Rewrite the integral with the completed substitution.
$\frac{{x}^{2}}{2}-4x+\frac{\mathrm{ln}\left({x}^{2}-5\right)}{2}$
Step 10: Add constant.
$\frac{{x}^{2}}{2}-4x+\frac{\mathrm{ln}\left({x}^{2}-5\right)}{2}+C$

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